= Azimuthal derivative of a cylindrical vector Fourier mode
{title2=$\partial_\phi\mathbf v=im\mathbf v+\widehat{\mathbf z}\times\mathbf v$}
If the cylindrical components of a <vector field> have dependence $e^{im\phi}$, differentiation of the rotating basis adds the displayed <cross product> term. Scalar-component differentiation alone misses this contribution. For $\mathbf B=(J/2)s\widehat{\boldsymbol\phi}$, one also has $(\mathbf v\cdot\nabla)\mathbf B=(J/2)\widehat{\mathbf z}\times\mathbf v$. These identities are useful in cylindrical <magnetohydrodynamic waves>.
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